Prime numbers: Decoding the DNA of maths

One of the leading Millennium Maths problems is 'the Riemann Hypothesis.' This problem relates to the Riemann Zeta function - a very complicated mathematical expression - but understanding it will tell us a lot more about prime numbers and how they are distributed. World famous mathematician Marcus du Sautoy, from Oxford University, joins Kat Arney to unpick the primes...

Marcus - Riemann was one of the greatest mathematicians of the 19th century in Germany. He was at one of the great hotbeds of mathematics Göttingen University, was a student of Carl Friedrich Gauss. He was also responsible for coming up with some of the mathematics that Einstein used in order to create relativity, the invention and discovery of dimensions of geometry beyond our 3-dimensional universe.

But the Riemann Hypothesis is great discovery about what we think is actually at the heart of making the primes tick. Interestingly, he discovered it in 1859, which is the same year as Charles Darwin's publication of the Origin of Species. I think you can call this really the origin of the primes. We think that what he discovered is telling us all about prime numbers.

But it's still something of a mystery. So, the Riemann Hypothesis, let me tell you what you've got to solve. You've to prove that the non-trivial zeros of the Riemann Zeta Function, that the real part of them all have a or are equal a half.

Kat - Hang on! So, I think that's maybe more maths than I can cope with right now. So, let's start with some simple stuff about what are the prime numbers about?

Marcus - A prime number probably you remember from school or maybe you're still at school. It's a number which is indivisible. You can't divide it by anything other than itself and one. So, it's something like 17 - the number I've played for on my football team - that's a prime number but 15 isn't because you're going to write that as 3 times 5.

Kat - Is there a pattern to how these prime numbers are distributed? We've got 3, 5, 7, 11, 13 and then it starts to get more spread out. Can we actually predict where they're going to be or is this random?

Marcus - Well, exactly. So, mathematics, I tend to think of, it's the science of patterns. That's what this is all about. These numbers, they're really the atoms of mathematics because you can build all other numbers by multiplying these primes together. But when you look at the list of them, 2, 3, 5, 7, 11, 13, and you count higher and higher, one of the first great theorems of the ancient Greeks was that they're in infinitely many of them. But then if they're meant to be many, we've got to try and find a pattern for this.

The chemists, they've got a periodic table which they just list all of their atoms but for the mathematicians, these prime numbers, the challenge is, there doesn't seem to be any patterns of these numbers which are the most fundamental in the whole of mathematics. So, that's the challenge.

It's really a 2000-year-old problem rather than a 150-year-old problem. But 150 years ago, Riemann actually understood that although outwardly they look very random, there's something hiding behind there called the Zeros of the Riemann Zeta Function which do have some pattern to them. And that's what you got to prove - a pattern about this kind of like the DNA which makes these numbers work.

Kat - So, how would someone go about proving this and then what could the payoff be? Why would it help if we could actually spot this underlying pattern and prove this to be true?

Marcus - If I knew the answer to your first problem, I wouldn't be here now. I'd be claiming my million dollars. So, that's one of the challenges that with these grey unsolved problems, how on earth do you go about solving them? Riemann actually stumbled on his new way of looking at the primes almost by chance. It's about playing and experimenting, trying to find connections with other things.

So, how are we going to do it? I don't know. Frankly, I'd pay a million dollars to solve this one. It's so important.

Why it will be important for people who perhaps or what's maths got to do with me? Actually, the prime numbers are absolutely essential now in all the codes that are used on the internet to send things securely. So every time you ever sent your credit card across the internet and want it to be kept secure, actually, you're using the fact that we don't understand the primes well enough to keep that secure. The code we use is prime numbers that are being code to things so it can't be read. If we understood the Riemann Hypothesis, potentially, the deeper understanding we'd have of primes would give us a way perhaps to crack these codes.

Kat - That sounds kind of risky to me though if all these information has been sent using prime numbers.

Marcus - Yes and potentially it is, but at the moment, the codes are very robust. But in a sense, it is tapping into the fact that we don't understand numbers. Here's a challenge - if I give you a number like 15, what are the primes which made that number?

Well, that's easy. It's 3 times 5. But if I give you a 200-digit number, which is the numbers used on the internet, to crack any website's cryptography, there's a public 200-digit number, you got to find the two primes which multiply together, give that number. So, if you can do that, it's kind of like a prime number spectrometer then you'll be able to crack all the codes on the internet.